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Question Number 128932 by Koyoooo last updated on 11/Jan/21

Commented by mr W last updated on 11/Jan/21

$$questioniswrong!$$$$-3⩽2a^2+b^2+2a-4b⩽44$$

Commented by liberty last updated on 12/Jan/21

$$\mathrm{yes}\mathrm{question}\mathrm{wrong}$$$$$$

Commented by liberty last updated on 12/Jan/21

$$0⩽\mathrm{a}⩽2\to \{\begin{array}{l}0⩽{2\mathrm{a}}^2⩽8\\ 0⩽2\mathrm{a}⩽4\end{array}\iff 0⩽{2\mathrm{a}}^2+2\mathrm{a}⩽12$$$$-4⩽\mathrm{b}⩽1\to \{\begin{array}{l}0⩽{\mathrm{b}}^2⩽16\\ -4⩽-4\mathrm{b}⩽16\end{array}\iff -4⩽{\mathrm{b}}^2-4\mathrm{b}⩽32$$$$\mathrm{but}\mathrm{b}\ne 2\mathrm{then}-3⩽{\mathrm{b}}^2-4\mathrm{b}⩽32$$$$\mathrm{adding}\left(1\right)\mathrm{and}\left(2\right)\Rightarrow -3⩽{2\mathrm{a}}^2+2\mathrm{a}+{\mathrm{b}}^2-4\mathrm{b}⩽44$$$$$$

Answered by mr W last updated on 11/Jan/21

$$2a^2+2a+b^2-4b$$$$=2(a^2+a+\frac{1}{4})+b^2-4b+4-\frac{1}{2}-4$$$$=2{(a+\frac{1}{2})}^2+{(b-2)}^2-\frac{9}{2}$$$$max=2{(2+\frac{1}{2})}^2+{(-4-2)}^2-\frac{9}{2}=44$$$$min=2{(0+\frac{1}{2})}^2+{(1-2)}^2-\frac{9}{2}=-3$$

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